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We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles. The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability. We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments. [ABSTRACT FROM AUTHOR]

The bounded functional interpretation of arithmetic in all finite types is able to interpret principles like weak König's lemma without the need of any form of bar recursion. This interpretation requires the use of intensional (rule-governed) majorizability relations. This is a somewhat unusual feature. The main purpose of this paper is to show that if the base domain of the natural numbers is extended with nonstandard elements, then the bounded functional interpretation can be seen as falling out from a functional interpretation of nonstandard number theory without intensional notions. The original bounded functional interpretation can be seen as the trace left behind by the new interpretation when one sees it restricted to the standard number theoretical setting. We also answer an open question regarding the conservativity of the transfer principle vis-à-vis functional interpretations of nonstandard arithmetic. [ABSTRACT FROM AUTHOR]